Null Hypothesis H_0: All of the k population distribution functions are identical.

Alternative Hypothesis H_A: At least one of the populations tends to yield larger observations than at least one of the other populations.

or

Alternative Hypothesis H_A: The k populations do not all have identical means. (k is the number of groups here.)

Rank sum is employed for hypotheses testing. Note that the sample size for each group do not have to be the same. Go to the Kruskal-Wallis Test Applet.

SOCR Analyses Example on the Kruskal-Wallis Test

The example we show here is from "Practical Nonparametric Statistics" by Conover, Second Edition, John Wiley & Sons, 1980. (Page 230)

Steps:

The data of this example has 4 groups: A, B, C and D. Click on "Kruskal-Wallis Test" from the conbo box in the left panel

Click on "Example 2" button. than the "Data" button to see the data

The data need to be send to the computer for analysis

In the Kruskal-Wallis test, at least two groups need to be included. You can certainly include any two groups, any three groups, or all the available groups, etc., from the data set. The test will only analyze the groups selected. Therefore, you'll need to let the computer know what groups you're choosing. This is done in the "Mapping" panel. Click on the "Mapping" button to include the groups you want.

Next, click on the "Calculate" button

Now you're ready to see the results. Just click on the "Result" button.

Scroll down to see complete results.

If you'd like to include some other group(s) or remove the current groups and start over, simply go to the Mapping button and take the groups you want.

Note: if you happen to click on the "Clear" button in the middle of the procedure, all the data will be cleared out. Simply start over from step 1 and retrieve the data by click an EXAMPLE button.

It is computed using T and S above. N is the total number of observations.

Notation: Ri -- Sum of ranks of group i; ni -- size of group i.

|Ri/ni - Rj/nj| 2.1199 * sqrt(24.9187) * sqrt(1/ni + 1/nj)

Group A vs. Group B: 3.1999 < 6.6928 (does not yield rejection of Ho)

Group A vs. Group C: 4.7000 < 6.6928 (does not yield rejection of Ho)

Group A vs. Group D: 5.7000 < 6.6928 (does not yield rejection of Ho)

Group B vs. Group C: 7.9 > 6.6928 (yields rejection of Ho)

Group B vs. Group D: 8.9 > 6.6928 (yields rejection of Ho)

Group C vs. Group D: 1.0 < 6.6928 (does not yield rejection of Ho)

Since Group B appears to have different mean ranks than group C and group D, the null hypothesis is rejected.

These results include the critical value (2.120) and the value of the test statistics (T=7.487), which indicate that there is sufficient evidence (at α = 0.05) to reject the null hypothesis that the sample centers are equal.

Also, the paired-group comparisons (accounting for multiple-testing) indicate the normal ranges of the rank differences between groups. For instance, Group B vs. Group C: 7.9 > 6.6928 indicates the range of normal rank differences between groups B and C.